qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
98,605 | <p>The <a href="http://en.wikipedia.org/wiki/Characteristic_%28algebra%29"><em>characteristic</em> of a ring</a> (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. <em>Otherwise, we define it to be $0$.</em> </p>
<p>Bu... | sdcvvc | 12,523 | <p>There are two orderings of the set $\mathbb N = \{0,1,\dots\}$:</p>
<ul>
<li>magnitude $a \leq b$</li>
<li>divisibility $a\mid b$ (i.e. $\exists c. b = a c$)</li>
</ul>
<p>They are mostly compatible - <em>usually</em> when $a \mid b$, it holds $a \leq b$.</p>
<p>Some definitions are phrased using "greater than" o... |
98,605 | <p>The <a href="http://en.wikipedia.org/wiki/Characteristic_%28algebra%29"><em>characteristic</em> of a ring</a> (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. <em>Otherwise, we define it to be $0$.</em> </p>
<p>Bu... | Allawonder | 145,126 | <p>One way to see why this definition is natural is to consider that if $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}}$$ never vanishes for any <em>positive</em> integer $n,$ then it makes sense to say that the ring has characteristic $0$ since it takes <em>no</em> number of units to vanish under its additive oper... |
98,605 | <p>The <a href="http://en.wikipedia.org/wiki/Characteristic_%28algebra%29"><em>characteristic</em> of a ring</a> (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. <em>Otherwise, we define it to be $0$.</em> </p>
<p>Bu... | KCd | 619 | <p>This is all just a convention, and I'm adding an answer <span class="math-container">$8$</span> years late because nobody else has pointed out yet that many years ago some people <em>did</em> use the term "characteristic <span class="math-container">$\infty$</span>". Schilling does this in his book "T... |
3,008,942 | <p>A fair 6-sided die is rolled 5 times and the result is recorded for each roll.</p>
<p>a) How many different sequences of results are possible?</p>
<p>b) Of the possible sequences of results, how many of them contain exactly 3 rolls of a 4?
<br/>
<br/>
<br/>
I'm pretty sure part (a) is <span class="math-container">... | drhab | 75,923 | <p>There are <span class="math-container">$\binom53$</span> ways to select <span class="math-container">$3$</span> rolls out of <span class="math-container">$5$</span> that are destined to be the rolls that give a <span class="math-container">$4$</span>.</p>
<p>For a selected roll there is <span class="math-container"... |
598,041 | <p>Prove the following inequality for each $x$:</p>
<p>$$|\sin x+7\cos x| \leq \sqrt{50}$$</p>
<p>My remark: It seems to me that the solution uses the fact that $1^2+7^2=50$.</p>
<p>Thanks in advance!</p>
| Kelvin Soh | 113,615 | <p>One of the ways is to use the addition formula: $R \sin (a+x) = (R\cos a) \sin x + (R\sin a) \cos x$. We let $R \cos a = 1$ and $R \sin a = 7$. Hence $50 = 1^2 + 7^2 = R^2 (\cos^2 + \sin^2) = R^2$ and $R = \sqrt{50}$. </p>
<p>$|\sin x + 7\cos x| = |\sqrt{50} \sin (a+x)| \leq \sqrt{50}$</p>
|
3,088,503 | <p>For <span class="math-container">$r\in [0,1)$</span> define
<span class="math-container">$$
I(r) = \int_0^{2\pi}\frac{\cos(t) - r}{1 - 2r\cos t + r^2}\,dt.
$$</span>
Numerical experiments hint at <span class="math-container">$I(r) = 0$</span> for all <span class="math-container">$r\in [0,1)$</span> but I can't show ... | omegadot | 128,913 | <p>Here is yet another slightly different approach.</p>
<p>Let
<span class="math-container">$$P_r(t) = \frac{\cos t - r}{1 - 2r \cos t + r^2}, \qquad r \in [0,1).$$</span>
Using <span class="math-container">$\cos t = (e^{it} + e^{-it})/2$</span>, we can rewrite <span class="math-container">$P_r(t)$</span> as
<span cla... |
3,088,503 | <p>For <span class="math-container">$r\in [0,1)$</span> define
<span class="math-container">$$
I(r) = \int_0^{2\pi}\frac{\cos(t) - r}{1 - 2r\cos t + r^2}\,dt.
$$</span>
Numerical experiments hint at <span class="math-container">$I(r) = 0$</span> for all <span class="math-container">$r\in [0,1)$</span> but I can't show ... | Ali Shadhar | 432,085 | <p>From <a href="https://math.stackexchange.com/q/3428840">the identity</a></p>
<p><span class="math-container">$$\sum_{n=1}^{\infty}r^{n-1} \cos(nt)=\frac{\cos(t)-r}{1-2r\cos(t)+r^2}, \ |r|<1$$</span></p>
<p>it follows that </p>
<p><span class="math-container">$$I(r)=\sum_{n=1}^\infty r^{n-1}\int_0^{2\pi}\cos(nt... |
3,621,135 | <p>Prove that the metric space <span class="math-container">$P[a,b]$</span> of all polynomials with uniform metric <span class="math-container">$d(f,g) = \max|f(t)- g(t)|$</span> for every <span class="math-container">$f, g$</span> belonging to <span class="math-container">$P[a,b]$</span> and <span class="math-containe... | jvc | 686,748 | <p>I suppose that <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are scalars (f there are matrix, they must commute in general).</p>
<p>The trick is to use the uniqueness in Cauchy Lipschitz theorem (wich gives the group property of the flow of an ODE). </p>
<p>Let <span class="m... |
124,694 | <p>I'm teaching an honors differential equations class and have been using linear algebra heavily. I thought it would be interesting to include a proof of the method of undetermined coefficients along the lines described below, which I figured is already known. Nonetheless, Google turns up no proof other than the obv... | Terry Tao | 766 | <ol>
<li><p>If $L$ has characteristic polynomial $\lambda \mapsto p(\lambda)$, then the conjugated differential operator $e^{-qt} L e^{qt}$ has characteristic polynomial $\lambda \mapsto p(\lambda+q)$. From this we may easily reduce the problem to the $q=0$ case.</p></li>
<li><p>If the characteristic polynomial $\lamb... |
2,270,401 | <p>I have heard about Euler bricks, and i have noticed that there are an <strong>strong text</strong>infinite number of euler bricks(<a href="https://math.stackexchange.com/questions/2264401/euler-bricks-and-the-4th-dimension">as for 4D bricks we are still pending even one</a>!) in existence. I get there may be infinit... | Koen Thomassen | 1,087,758 | <p>Among others, Euler and Saunderson devised ways to generate an infinitude of euler bricks, eg Saunderson starts with a pythagorean triple (u,v,w) and sets a=u|4v^2-w^2|, b=v|4u^2-w^2|, c=4uvw.
However these methods do not generate all (primitive) Euler bricks, so we cannot use them to prove the nonexistence of a per... |
103,603 | <p>There is a lock with the password between 000-999. But we could use only two digits to open the lock. For example, if the password is 123, the lock is open when you try 1*3, or *23, or 12*.
questions:
a) compose a strategy that the try number of opening the lock is the smallest one under the worst situation
b) Prov... | John | 23,964 | <p>What I think is a strategy like this: </p>
<p>0 1 X (X=0,1...9) - 10 times</p>
<p>2 3 X (X=2,3...9) - 8 times</p>
<p>4 5 X (X=4,5...9) - 6 times </p>
<p>6 7 X (X=6,7,8,9) - 4 times </p>
<p>8 9 X (X=8,9) - 2 times</p>
<p>This strategy could cover all digits combinations, including the duplicated d... |
1,529,885 | <p>I have $24$ items randomly divided into $4$ groups (it can happen that group has $0$ items). How can I prove that when I remove a maximum of $12$ items (of my choice) three boxes will have the same amount?</p>
| JMP | 210,189 | <p>The conjecture is false, there are two counterexamples:</p>
<p>$$0, 3, 10, 11$$</p>
<p>and</p>
<p>$$0, 2, 11, 11$$</p>
|
1,990,274 | <p>The problem asks:</p>
<blockquote>
<p>A silo (base not included) is to be constructed in the form of a
cylinder surmounted by a hemisphere. The cost of construction per
square unit of surface area is $7$ times as great for the hemisphere
as it is for the cylindrical sidewall. Determine the dimensions to b... | देवेन्द्र prasad | 222,805 | <p>A very good lecture series <a href="https://www.youtube.com/playlist?list=PLbMVogVj5nJRR7zYZifYopb52zjoScx1d" rel="nofollow noreferrer">can befound here</a></p>
|
164,114 | <p>Is a commutative ring a field? A set equipped with addition and multiplication which is abelian over those two operations and it holds distributivity of multiplication over addition?</p>
| Community | -1 | <p>A key difference between an ordinary commutative ring and a field is that in a field, all non-zero elements <strong><em>must</em></strong> be invertible. For example:</p>
<p>$\Bbb{Z}$ is a <strong><em>commutative ring</em></strong> but $2$ is not invertible in there so it can't be a field, whereas $\Bbb{Q}$ is a <s... |
3,602,660 | <p>More clearly the question is:</p>
<p>Prove that R is a field by showing that <span class="math-container">$\mathbb{R}$</span> forms an isomorphisim to <span class="math-container">$\mathbb{C}$</span> as rings (explicitly find an isomorphism between them and prove it is an isomorphism).</p>
<p>Let <span class="math... | Eric Towers | 123,905 | <p>First,
<span class="math-container">$$ (x-0)(x-0)(x-0) = x^3 $$</span>
exhibits the three roots of <span class="math-container">$x^3$</span>. They're all <span class="math-container">$0$</span>.</p>
<p>Much like the discriminant of the quadratic, the <a href="https://en.wikipedia.org/wiki/Discriminant#Degree_3"... |
2,913,425 | <p>In </p>
<p>E. Swenson, A cut point theorem for CAT(0) groups, J. Differential Geom. 53 (1999), no. 2, 327–358. </p>
<p>the author shows (Theorem 11) that if a group $G$ acts geometrically (i.e. properly discontinuously) and cocompactly by isometries on a CAT(0) space, then $G$ has an element of infinite order. How... | Steve D | 265,452 | <p>It seems the proof is already assuming the group is infinite. Certainly it is true for infinite groups (this is well-known). Maybe somewhere the author mentions he's only talking about infinite groups?</p>
|
98,971 | <p>I'm working on an algorithm which should check if two polygons, described by their vertex coordinates, are: one inside the other, are intersecting or are separated</p>
<p>image below describe this three cases:
<a href="https://i.stack.imgur.com/hl6zL.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.co... | paw | 4,539 | <p>Using <code>RegionIntersection</code> and <code>Area</code>:</p>
<pre><code>PolygonIntersectingQ[poly1_, poly2_] := Module[{m1, m2, area},
{m1, m2} = MeshRegion[#, Polygon[Range@Length@#]] & /@ {poly1, poly2};
area = Area@RegionIntersection[m1, m2];
Switch[area,
0, False,
Area@m1, poly1,
Ar... |
2,794,717 | <p>I am stuck on the following question.</p>
<p>Prove that $S(n+1, n) = \frac{n(n+1)}{2}$ for any $n \ge 1$</p>
<p>This is where $S(n+1,n)$ is the Stirling number, partitioning a $n+1$-element set into $n$ parts.</p>
<p>What first steps should be taken to try and derive this?</p>
| Angina Seng | 436,618 | <p>Let's suppose you have a set of eleven distinguishable balls and ten blank
boxes to put them in. Also you are commanded to put at least one ball in each box. Then you have to have one box with two balls and nine boxes
with one ball each. You can distinguish the two-ball box now, but not the
one-ball boxes. So you ca... |
4,521,067 | <blockquote>
<p>If <span class="math-container">$$f(x)=\prod_{k=1}^\infty\left(1-\frac{x^3}{k^3\pi^3}\right)$$</span>
and if <span class="math-container">$$f(x)=\sum_{n=0}^\infty a_n x^n$$</span>
Then find <span class="math-container">$a_n$</span> or convert the above infinite product into an infinite sum.</p>
</block... | GEdgar | 442 | <p>remark.
If
<span class="math-container">$$
\prod_{k=1}^\infty\left(1-\frac{x^3}{k^3\pi^3}\right)
=\sum_{n=0}^\infty a_n x^n
$$</span>
then
<span class="math-container">\begin{align}
a_0 &= 1 ;\\
a_3 &= \sum_{k=1}^\infty \frac{-1}{k^3\pi^3} = -\frac{\zeta(3)}{\pi^3}
\approx -0.03876817958 ;\\
a_6 &= \sum_... |
3,514,934 | <blockquote>
<p>If <span class="math-container">$x = 2 + \sqrt{3}$</span> then find <span class="math-container">$x^3 + \dfrac1{x^3}$</span>. </p>
</blockquote>
<p>My attempt: we know that the sum of a number and its reciprocal is always greater than or equal to <span class="math-container">$2$</span> but I don't kn... | Michael Rozenberg | 190,319 | <p><span class="math-container">$$(2+\sqrt3)^3+\frac{1}{(2+\sqrt3)^3}=(2+\sqrt3)^3+(2-\sqrt3)^3=2\cdot2^3+2\cdot3\cdot2\cdot\left(\sqrt3\right)^2=52.$$</span></p>
|
543,531 | <p>Suppose $X$ and $Z$ are standard normal random variables and define $$Y = X + Z.$$
Then $Y$ is also normal, has zero mean and its variance is two (<a href="http://en.wikipedia.org/wiki/Normal_distribution#Related_distributions" rel="nofollow">Wikipedia</a>). It should also be fair to assert that $Y|(X=x)$ has a norm... | r.e.s. | 16,397 | <blockquote>
<p>Suppose $X$ and $Z$ are standard normal random variables and define $$Y = X + Z.$$
Then $Y$ is also normal, has zero mean and its variance is two (<a href="http://en.wikipedia.org/wiki/Normal_distribution#Related_distributions" rel="nofollow noreferrer">Wikipedia</a>).</p>
</blockquote>
<p>You're f... |
61,513 | <p>Hello. I'd like to consider the open unit ball in an infinite dimensional Hilbert space and ask when can we fit infinitely many open balls of radius $r<1$ inside.</p>
<p>For example, when $r=1/(1+\sqrt2)$, we can pick an orthonormal basis $(x_1,...)$ for our Hilbert space and put the centers of the balls at $(1-... | Mikael de la Salle | 10,265 | <p>Your value of $r$ is the best.</p>
<p>Equivalently, $\rho=\sqrt 2$, where $\rho$ is the sup, over all infinite sequence $(x_i)$ in the unit ball of a Hilbert space, of $\inf_{i\neq j} |x_i-x_j|$.</p>
<p>Here is a proof, by contradiction. Assume that $\rho>\sqrt 2$ and pick a sequence $(x_i)$ such that $\inf_{i\... |
1,632,879 | <p>$$\lim_{n\to \infty}\sum_{r=1}^n \tan^{-1} \dfrac{2r+1}{r^4+2r^3+r^2+1}$$</p>
<p>How am I supposed to do it? One thing I see here is </p>
<p>$$\lim_{n\to \infty}\sum_{r=1}^n \tan^{-1} \dfrac{2r+1}{(r^2+r)^2+1}$$</p>
<p>Here derivative of $r^2+r$ is $2r+1$. (if it helps)</p>
<p>It's final answer is</p>
<blockquo... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>Since $$\dfrac{2r+1}{(r^2+r)^2+1}=\dfrac{(r+1)^2-r^2}{1+r^2(r+1)^2}$$ then
$$\tan^{-1} \Big(\dfrac{2r+1}{(r^2+r)^2+1}\Big)= \tan^{-1}(1+r)^2- \tan^{-1}(r^2) $$ which beautifully telecospes.</p>
|
3,527,470 | <p>Compute <span class="math-container">$$\lim\limits_{n\to \infty} \int\limits_0^1 x^{2019} \{nx\} dx,$$</span> where <span class="math-container">$\{a\}$</span> denotes the fractional part of the real number <span class="math-container">$a$</span>.<br>
I firstly tried to apply the substitution <span class="math-conta... | Paramanand Singh | 72,031 | <p>Here is another approach which is somewhat simpler than the one given in another answer here.</p>
<p>I establish that <span class="math-container">$$\int_{0}^{1}f(x)\{nx\}\,dx\to\frac{1}{2}\int_{0}^{1}f(x)\,dx$$</span> as <span class="math-container">$n\to\infty $</span>. The integral on left of above equation can ... |
12,006 | <p>Recently I was asked to tutor some students individually on a basic course of Linear Programming, centered on all the "typical" introductory subjects like graphical methods, Simplex Algorithm and Transfer Pricing.</p>
<p>Now, this is not the first time I teach someone on a University level, but it is the first time... | aparente001 | 4,956 | <p>Tell the person who asked you that you are willing to do some extra preparation on your own to carry out this assignment, but frankly they'd be better off giving this assignment to someone else, because you and linear programming didn't "click" when you took the class.</p>
<p>If they can't find anyone else, they'll... |
1,386,549 | <p>What is the range of $x+\frac{1}{x}$ if $-2\leq x \leq 2$?<br></p>
<p>My try:<br></p>
<p>$-2\leq x \leq 2$<br></p>
<p>$\Rightarrow \frac{-1}{2}\leq x \leq \frac{1}{2}$<br></p>
<p>$\Rightarrow \frac{-5}{2}\leq x+\frac{1}{x} \leq \frac{5}{2}$<br></p>
<p>Am i correct or wrong?If not,please help me solve this.</p>
| Elliott de Launay | 584,468 | <p>The textbook answer says that the following represents:</p>
<blockquote>
<p>Some student has asked every faculty member a question.</p>
</blockquote>
<p><span class="math-container">$\forall y(F(y) \to \exists x (S(x) \land A(x,y))) $</span></p>
<p>Which seems incorrect as I read this as saying for every facul... |
27,968 | <p>What is the "standard" way to denote all positive (or non-negative) real numbers? I'd think</p>
<p>$$ \mathbb R^+ $$</p>
<p>but I believe that that is usually used to denote "all real numbers including infinity".</p>
<p>So is there a standard way to denote the set</p>
<p>$$ \{x \in \mathbb R : x \geq 0\} \; ?$$ ... | hwong557 | 8,368 | <p>Some of my profs use $\mathbb{R^{\ge 0}}$. I like to add whatever to the top so $\mathbb{R^{\le a}}$ just means all reals less than $a$.</p>
|
27,968 | <p>What is the "standard" way to denote all positive (or non-negative) real numbers? I'd think</p>
<p>$$ \mathbb R^+ $$</p>
<p>but I believe that that is usually used to denote "all real numbers including infinity".</p>
<p>So is there a standard way to denote the set</p>
<p>$$ \{x \in \mathbb R : x \geq 0\} \; ?$$ ... | D Left Adjoint to U | 26,327 | <p>$\mathbb{R}^+$ includes $0$ in <a href="http://www.probability.net" rel="nofollow">Probability Tutorials</a>. $\mathbb{R}^+_0$ is more clear though, so I've used <em>it</em> in the exercises.</p>
|
27,968 | <p>What is the "standard" way to denote all positive (or non-negative) real numbers? I'd think</p>
<p>$$ \mathbb R^+ $$</p>
<p>but I believe that that is usually used to denote "all real numbers including infinity".</p>
<p>So is there a standard way to denote the set</p>
<p>$$ \{x \in \mathbb R : x \geq 0\} \; ?$$ ... | Patrick | 109,962 | <p>I find <span class="math-container">$\mathbb R_{\geq 0}$</span> clumsy (I would never write this on a board when working and I don't often see papers with functions <span class="math-container">$f:\mathbb R_{\geq 0}\rightarrow \mathbb R_{\geq 0}$</span>).</p>
<p><span class="math-container">$\mathbb R^+$</span> seem... |
450,115 | <p>The Wikipedia page of <a href="http://en.wikipedia.org/wiki/Ackermann_function">Ackermann function</a> states that Ackermann function is "roughly comparable" to $f_\omega$ in <a href="http://en.wikipedia.org/wiki/Fast-growing_hierarchy">fast-growing hierarchy</a>. </p>
<p>Is there some standard way to make the "rou... | Simply Beautiful Art | 272,831 | <p>Generally, when one compares functions to the fast growing hierarchy, we say that a function $g(n)$ grows at $f_\alpha(n)$ if $g(n)$ grows faster than $f_\beta(n)$ for every $\beta<\alpha$ and $g(n)$ grows slower than $f_\alpha^k(n)$ for a fixed $k$. For functions/notations that come with multiple arguments, we ... |
42,624 | <p>In the <a href="http://arxiv.org/abs/math/0606464v1">http://arxiv.org/abs/math/0606464v1</a> I read </p>
<p>"If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in
possession of a knot which is topologically slice but not smoothly slice (slice means zero slice genu... | Tim Perutz | 2,356 | <p>From Jacob Rasmussen's paper "Knot polynomials and knot homologies", arXiv:math/0504045, p.13 of ArXiv version:</p>
<blockquote>
Bob Gompf kindly pointed out another such application [of Rasmussen's $s$-invariant, a concordance invariant of knots extracted from Khovanov homology]. Namely, $s$ can also be used to g... |
2,295,231 | <p>Let $A=(a_{ij}) $ be a $2\times 2$ real matrix such that $a_{11}^2+a_{22}^2+a_{21}^2+a_{12}^2 < 1/10 $. </p>
<p>Prove that $I+A$ is invertible .</p>
<p>My approach : Suppose $I+A$ were singular . Then it has a non-zero $2\times 1$ column vector such that $(I+A)v=0 $ i.e $Av=-v $. Hence $-1$ is an eigenvalue of ... | Robert Z | 299,698 | <p>I think we can get a simpler proof. Note that the inequality
$$a_{11}^2+a_{22}^2+a_{21}^2+a_{12}^2 < \frac{1}{10}$$
implies that $|a_{ij}|< \frac{1}{\sqrt{10}}$. Hence
\begin{align*}|\det(A+I)|&=|1+a_{11}+a_{22}+a_{11}a_{22}-a_{21}a_{12}|\\
&\geq 1-|a_{11}|-|a_{22}|-|a_{11}||a_{22}|-|a_{21}||a_{12}|\\
... |
2,157,250 | <p>Exponential Blowup of Subset Construction
Definition: $L_n = \{w ∈ \{0, 1\}^\star\,|\,w_{|w|-n}=1\}$. In other words,
$L_n$ is the set of binary words having a $1$ in the nth position from the end. Here $|w|$ is the length of the word $w$.</p>
<p>Note: an NFA with only $n+1$ states can be designed to recognized $L_... | Adam Hughes | 58,831 | <p>Write this as</p>
<blockquote>
<p>$$\begin{pmatrix}
a && 1 \\
1 && -b
\end{pmatrix}\begin{pmatrix}x \\ 1\end{pmatrix}$$</p>
</blockquote>
<p>Then this is self inverse iff</p>
<p>$$\begin{pmatrix}
a && 1 \\
1 && -b
\end{pmatrix}^2= \lambda I_2$$</p>
<p>i.e.</p>
<p>$$\begin{pmatr... |
2,157,250 | <p>Exponential Blowup of Subset Construction
Definition: $L_n = \{w ∈ \{0, 1\}^\star\,|\,w_{|w|-n}=1\}$. In other words,
$L_n$ is the set of binary words having a $1$ in the nth position from the end. Here $|w|$ is the length of the word $w$.</p>
<p>Note: an NFA with only $n+1$ states can be designed to recognized $L_... | Fimpellizzeri | 173,410 | <p>Suppose $b\neq 0$. Then $f(0)=-1/b$ and if $f$ is self inverse</p>
<p>$$0=f(-1/b)=\frac{a/b-1}{b+1/b}=\frac{a-b}{b^2+1}.$$</p>
<p>It follows that $a=b$. Then</p>
<p>$$f(f(x))=\frac{a\frac{ax+1}{x-a}+1}{\frac{ax+1}{x-a}-a}=\frac{a^2x+a+(x-a)}{ax+1-a(x-a)}=\frac{a^2x+x}{1+a^2}=x$$</p>
<p>and $f$ is self-inverse. I... |
4,087,664 | <p>Solve the inequality <span class="math-container">$\frac{7 - |2x+1|}{2\cos(x)-7} < 0 $</span>.<br />
I have no clue how to solve this inequality. Please give a hint.</p>
| Aryaman Maithani | 427,810 | <p>You are looking for all those elements which whose (monic) irreducible polynomial over <span class="math-container">$\Bbb F_2$</span> has degree exactly <span class="math-container">$4$</span>.</p>
<p>We have an explicit formula to get the number. This is the <a href="https://en.wikipedia.org/wiki/Necklace_polynomia... |
1,221,795 | <blockquote>
<p>Suppose $f$ is continuous on $[a, b]$, if for every continuous function $g$ on $[a, b]$ with $g(a) = g(b) = 0, \int_{a}^{b}f(x)g(x) dx = 0$, Show $f(x) = 0, \forall x \in [a, b]$, </p>
</blockquote>
<p>I want to prove by contradiction, and then find a continuous $g$ such that $g(a) = g(b) = 0$ but $... | r12 | 10,140 | <p>Suppose there exists some $n$ such that $f(n) = a$, $a \neq 0$. Since $f$ is continuous, let $E$ be the open interval around $n$ where $E \subset [a,b] $. </p>
<p>Let $E1$ be the open interval around $n$ such that $sgn(e) = sgn(n) \forall e\in E1 \subset E$.</p>
<p>Let $E2 \subset E1$.</p>
<p>Let $E2 = (c_2, d_2... |
795,991 | <p>What I'm really wondering is, does Lagrange polynomial interpolation have an answer for every question of "what's the next integer in this sequence"? Does it define an infinite integer sequence to extend every finite integer sequence?</p>
<p>Let $S$ be a finite set of points, at least two of them, all having consec... | Sai Teja | 489,011 | <p>This is an exact differential equation.
$\frac{\partial}{\partial x} (2x^3y−3xy^2+1) = \frac{\partial}{\partial y} (3x^2y^2−y^3+2x) = 6x^2y-3y^2$</p>
<p>Answer is straightforward:
$$x^3y^2-xy^3+y+x^2 = c$$</p>
<p>Look up "Exact differential equations", if you still have a problem with these type of questions</p>
|
3,372,755 | <p><span class="math-container">$w$</span> and <span class="math-container">$z$</span> are given in polar form as <span class="math-container">$w=Re^{i\Phi}$</span>, <span class="math-container">$z=re^{i\theta}$</span>, <span class="math-container">$R>r>=0$</span>;</p>
<p>I can prove this using decomposition int... | Z Ahmed | 671,540 | <p><span class="math-container">$$\frac{w+z}{w-z}=\frac{(w+z)(\bar{w}-\bar{z})}{(w-z)(\bar{w}-\bar{z})}$$</span> <span class="math-container">$$=\frac{|w|^2-|z|^2-(\bar{z}w-z\bar{w})}{|w-z|^2}.$$</span> Here <span class="math-container">$(\bar{z}w -z \bar{w})$</span> is purely imaginary.</p>
<p><span class="math-conta... |
3,372,755 | <p><span class="math-container">$w$</span> and <span class="math-container">$z$</span> are given in polar form as <span class="math-container">$w=Re^{i\Phi}$</span>, <span class="math-container">$z=re^{i\theta}$</span>, <span class="math-container">$R>r>=0$</span>;</p>
<p>I can prove this using decomposition int... | sango | 346,515 | <p>Well, i you use <span class="math-container">$w=a+bi, z=c+di$</span> you can solve your problem operating.</p>
<p><span class="math-container">$\begin{equation}
\begin{aligned}
\frac{w+z}{w-z} &= \frac{(a+c)+(b+d)i}{(a-c)+(b-d)i} \\
& = \frac{(a+c)+(b+d)i}{(a-c)^2+(b-d)^2}\cdot((a-c)-(b-d)i) \\
& = \fra... |
1,554,108 | <p>Suppose $f$ is a nonnegative measurable function on a measure space $(X,M,\mu)$ satisfying $\int f d\mu < \infty$. Show that for every $\epsilon > 0 $ there exists a $\delta > 0 $ such that if $\mu(E) < \delta$ then $$\int_{E}f d\mu < \epsilon$$</p>
<p>Attempted proof: Let $E_n = \{f \geq 1/n\}$, the... | Robert Israel | 8,508 | <p>Work backwards. Find a vector $(0,y,z)$ in the YZ plane that can be rotated to $(1/\sqrt{3}, 1/\sqrt{3}, 1/\sqrt{3})$ by your second rotation (for the appropriate $\alpha$). Then find $\theta$ so your first rotation takes
$(0,0,1)$ to that vector. $\alpha$ and $\theta$ will not both be 45 degrees.</p>
|
629,923 | <p>Let $G$ be a finite solvable group and let $x_1,x_2,...,x_n$ be elements of $G$ of pairwise relatively prime orders. If the product $x_1x_2...x_n=1$, prove that each $x_i=1$.</p>
<p>I have no idea. Tell me some hints.</p>
<p>Thanks in advanced.</p>
| Robert Israel | 8,508 | <p>Hint: Use <a href="http://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow" title="Stirling's approximation">Stirling's approximation</a> to get the asymptotics of $n!$</p>
|
18,231 | <p>I'm stuck on the following problem :</p>
<p>Does there exist a real borel measure $\mu$ on $[0,1]$ such that</p>
<p>$$\int_{0}^{1}x^n d\mu = e^{-n^2}$$
for all $n \geq 1$ ?</p>
<p>Does anyone have any hint?</p>
<p>Thank you</p>
| NebulousReveal | 2,548 | <p>Here are some possible resources for problem solving:</p>
<ul>
<li><a href="http://rads.stackoverflow.com/amzn/click/0387982191" rel="nofollow noreferrer">Problem Solving Strategies</a> by Arthur Engel</li>
<li><a href="http://www.imo-official.org/problems.aspx" rel="nofollow noreferrer">Old IMO Problems</a></li>
<... |
18,231 | <p>I'm stuck on the following problem :</p>
<p>Does there exist a real borel measure $\mu$ on $[0,1]$ such that</p>
<p>$$\int_{0}^{1}x^n d\mu = e^{-n^2}$$
for all $n \geq 1$ ?</p>
<p>Does anyone have any hint?</p>
<p>Thank you</p>
| roy smith | 5,211 | <p>Read harder books, like Courant's or Spivak's or Apostol's Calculus, and Van der Waerden or M. Artin on algebra and linear algebra. If those are too easy try Spivak's Calculus on manifolds or Fleming's Calculus of several variables, or Dieudonne's Foundations of modern analysis and Jacobson's Basic algebra I, or Ch... |
1,682,306 | <p>I am looking to prove, by induction, the following equality:$${n \choose k} = {n \choose k-1}\frac{n-k+1}{k}$$ From Pascal's identity, I know we have that $${n+1 \choose k} = {n \choose k} + {n \choose k-1}$$ The problem gives the following hint: get ${n \choose k-1}$ and ${n+1 \choose k-1}$ in terms of ${n \choose ... | Aditya Dev | 249,297 | <p><strong>Using your method</strong>, $$\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}$$
Use $\binom{n+1}{k}=\frac{n+1}{k}\binom{n}{k-1}$
$$\frac{n+1}{k}\binom{n}{k-1}= \binom{n}{k}+\binom{n}{k-1}$$
$$\binom{n}{k}=\left(\frac{n+1}{k}-1\right)\binom{n}{k-1}=\frac{n-k+1}{k}\binom{n}{k-1}$$</p>
|
3,664,875 | <p>Is the ring <span class="math-container">$\mathbb{Z[x]}/(2x^2 + 5)$</span> isomorphic to any subring of <span class="math-container">$\mathbb{Z[x]}/(x^3+x+1)$</span>? I believe that the answer is No, but I am not 100 % sure. Here is my reasoning: </p>
<p><span class="math-container">$\mathbb{Z[x]}/(2x^2 + 5) = \{b ... | user350031 | 350,031 | <p>Maybe the following argument is along the lines you meant in your original post...</p>
<p>(Though the argument outlined in the other answer is obviously much simpler!)</p>
<hr>
<p>Let <span class="math-container">$S=\mathbb{Z}[X]/(X^3+X+1)$</span> and let <span class="math-container">$x=X+(X^3+X+1) \in S.$</span>... |
2,118,974 | <p>Prove that:$$\frac {\sec (16A) - 1}{\sec (8A) - 1}=\frac {\tan (16A)}{\tan (4A)}$$.</p>
<p>My Attempt,</p>
<p>$$L.H.S= \frac {\sec (16A)-1}{\sec (8A)-1}$$
$$=\frac {\frac {1}{\cos (16A)} -1}{\frac {1}{\cos (8A)} -1}$$
$$=\frac {(1-\cos (16A)).(\cos (8A)}{(\cos (16A))(1-\cos (8A))}$$.</p>
<p>What should I do next?... | juantheron | 14,311 | <p>$\displaystyle \frac{1-\cos^2(16 A)}{1+\cos (16A)}\times \frac{\cos 8 A}{\cos (16 A)} \times \frac{1+\cos 8A}{1-\cos^2 (8A)} = \frac{\sin^2 16A}{2\cos^2(8A)} \times \frac{\cos 8A}{\cos 16A} \times \frac{2\cos^2 4A}{\sin^2 8A}$</p>
<p>So $$ = \frac{\sin^2 (16 A)}{\sin (16A)} \times \frac{2\cos^2 (4A)}{\cos 16 A}\tim... |
982,582 | <p>Assume that $W$ and $V$ are two subspace of $X$. Is their union a subspace of $X$ too? I think it is not true unless under certain conditions but I do not know what conditions...</p>
| Milly | 182,459 | <p>No, union is not vector subspace in general (e.g. $\{y=x\}\cup \{y=-x\}$ in $\mathbb{R^2}$... direct sum $W\oplus V$ is a vector subspace).</p>
|
1,676,491 | <p>$\lim_{(x,y)\rightarrow 0}\frac{xy^2}{x^2+y^4}$</p>
<p>if we consider the path $x=y^2$ and get
$\lim_{(y^2,y)\rightarrow 0}\frac{y^2(y^2)}{(y^2)^2+y^4}=\lim_{y\rightarrow 0}\frac{y^4}{2y^4}=\frac{1}{2}$</p>
<p>but if I used Mathematrica to draw a graph of this function
<a href="https://i.stack.imgur.com/D6VWA.png"... | John Wayland Bales | 246,513 | <p>Since the math is impeccable I would suspect the fault lies with the values of $\Delta x$ and $\Delta y$ used by Mathematica. It has been years since I used Mathematica so I cannot advise you here. But if possible, set $\Delta x$ and $\Delta y$ to smaller values and see if you obtain a similar result.</p>
|
533,720 | <p>Let $A$ be open in $\mathbb{R}^m$; let $g:A\rightarrow\mathbb{R}^n$. If $S\subseteq A$, we say that $S$ satisfies the <strong>Lipschitz condition</strong> on $S$ if the function $\lambda(x,y)=|g(x)-g(y)|/|x-y|$ is bounded for $x\neq y\in S$. We say that $g$ is <strong>locally Lipschitz</strong> if each point of $A$ ... | Post No Bulls | 111,742 | <p>The Euclidean norm $f(x)=\|x\|$ is probably the simplest example: the Lipschitz condition comes straight from the triangle inequality, and the non-differentiability is seen from the fact that the restriction of $f$ to the first coordinate axis is $|x_1|$.</p>
|
583,544 | <p>Sometimes I see an exercise like this: </p>
<blockquote>
<p>Let $f:[a,b] \to \mathbb R$ be differentiable. (A few more givens here.) Show that $f'$ has such-and-such property.</p>
</blockquote>
<p>What is usually meant by that? Should $f'$ be defined on $[a,b]$ with one-sided derivatives in $a,b$? Or should one ... | coffeemath | 30,316 | <p>The closed form expression for the partial sum $S_n=\sum_{k=2}^n H_k/(k(k-1)$ is
$$S_n=2-\frac{1}{n}-\frac{H_n}{n}.$$
This can be shown by induction, or more easily by writing out the sum and using that $1/(k(k-1)=1/(k-1)-1/k$ to collect the contributions of $1,1/2,1/3,\cdots,1/n$ to the sum. As others have pointed... |
583,544 | <p>Sometimes I see an exercise like this: </p>
<blockquote>
<p>Let $f:[a,b] \to \mathbb R$ be differentiable. (A few more givens here.) Show that $f'$ has such-and-such property.</p>
</blockquote>
<p>What is usually meant by that? Should $f'$ be defined on $[a,b]$ with one-sided derivatives in $a,b$? Or should one ... | Ali Shadhar | 432,085 | <p><span class="math-container">$$\sum_{k=2}^{\infty} \frac{H_k}{k(k-1)}=\sum_{k=1}^{\infty} \frac{H_{k+1}}{k(k+1)}=-\int_0^1 \ln(1-x)\sum_{k=1}^\infty \frac{x^k}{k}dx$$</span></p>
<p><span class="math-container">$$=\int_0^1\ln^2(1-x)dx=\int_0^1 \ln^2xdx=2$$</span></p>
<p>Note that we use <span class="math-container">$... |
1,936,963 | <p>Let all other solutions diverge from $y = \frac{3}{4}$. write a differential equation $(\frac{dy}{dt} = ay +b)$ whose solutions have the required behavior as t goes to infinity. Explain how you came up with the differential equation).</p>
<p>I think the answer is $y' = 4y - 3$ but I am not sure. Can someone help me... | Jean-Claude Arbaut | 43,608 | <p>Let's write this $x_{n+1}=\dfrac12\left(x_n+\dfrac2{x_n}\right)$. The sequence $x_n$ has actually a <em>closed form</em>.</p>
<p>First, notice that $x_n>\sqrt{2}$ for all $n$, by induction: if $x_n>\sqrt{2}$, then</p>
<p>$$x_{n+1}-\sqrt2=\dfrac12\left(x_n-2\sqrt{2}+\dfrac2{x_n}\right)=\frac12\left(\sqrt{x_n}... |
1,886,332 | <p>Let $f$ be a continuous function from the reals $\mathbb{R}$ onto $I=[0,1]$ with usual topology. Prove that if $C$ is a subset of $I$ and the preimage of $C$ is closed in $\mathbb{R}$ then $C$ is closed in $I$.</p>
<p>My attempt is to use normal space properties but it does not help. </p>
| mb- | 356,869 | <p>Hint: it will be easier (in my opinion) to show that $C^c$ must be open in I, take a point in $x \in C^c$, and look for $r>0$ s.t $B(x,r) \subset C^c$. </p>
|
1,815,431 | <p>I'm currently trying to work out a function where</p>
<p>f(x) = 2x + 4</p>
<p>The question is to define f(x) <strong><em>recursively</em></strong></p>
<p>I'm unsure what this means, I have an idea of how to do it but I'm not sure if it's right. The way I thought of doing it was:</p>
<p>f(0) = 2*0 + 4 = 4<br>
f(1... | user341502 | 341,502 | <p>It's clear that each number in the sequence $4,6,8,10...$ is just the previous number increased by 2.</p>
<p>So we may write
$f(n) = 2+ f(n-1)$, when $n$ is a positive integer.
And $f(0)=4$.</p>
|
416,704 | <p><span class="math-container">$\newcommand{\R}{\mathbb{R}}$</span>I have a map <span class="math-container">$f\in C^0(X,\mathbb{R})$</span>, where <span class="math-container">$X$</span> is a compact and Hausdorff topological space, which is a manifold outside of a compact subset <span class="math-container">$K\subs... | Tim Campion | 2,362 | <p>EDIT:</p>
<p>In case you missed it, there's been a huge breakthrough in this area <a href="https://arxiv.org/abs/2203.14787" rel="nofollow noreferrer">by Robert Burklund</a>.</p>
<hr />
<p>I believe that <a href="https://arxiv.org/abs/1607.02702" rel="nofollow noreferrer">Prasit Bhattacharya's methods</a>, as linked... |
2,523,266 | <p>Find the value of $$\sum_{r=1} ^\infty \frac{r^3+(r^2+1)^2}{(r^4+r^2+1)(r^2+r)}$$
I tried converting the general term into partial fractions but the $$(r^2+1)^2$$ in the numerator is creating problems to do the sum. Please can somebody explain how can I compute the infinite sum</p>
| Piquito | 219,998 | <p>Each time that $(x+y\sqrt2)(x-y\sqrt2)=\pm1\iff x^2-2y^2=\pm1$ you have a unit in $Z[\sqrt2]$.</p>
<p>See about Pell-Fermat equation for details. In your particular case all the solutions are given by $$x_n+y_n\sqrt2= (1+\sqrt2)^n$$ For each $n\in\mathbb N$ you have a unit $x_n+y_n\sqrt2$.</p>
|
4,560,406 | <p>I am trying to model the temperature function using the following equation:</p>
<p><span class="math-container">$T(d)=c_0+c_1 \cos (\frac{2\pi}{365} d)$</span></p>
<p>Where <span class="math-container">$d$</span> is the day of the year, and <span class="math-container">$T(d)$</span> is the temperature on that day. I... | Zieac | 991,422 | <p>You may find <a href="https://en.wikipedia.org/wiki/Menelaus%27s_theorem" rel="nofollow noreferrer">Menelaus Theorem</a> useful, as well as the proof.
According to your description, the line <span class="math-container">$EC$</span> cut through three edges of the triangle <span class="math-container">$ABD$</span> at ... |
4,560,406 | <p>I am trying to model the temperature function using the following equation:</p>
<p><span class="math-container">$T(d)=c_0+c_1 \cos (\frac{2\pi}{365} d)$</span></p>
<p>Where <span class="math-container">$d$</span> is the day of the year, and <span class="math-container">$T(d)$</span> is the temperature on that day. I... | HackR | 1,088,726 | <p>Here's a cute little trick by using the uniqueness of center of mass.</p>
<p>Since we know that <span class="math-container">$AD:DC=1:2$</span>, place a mass of <span class="math-container">$2$</span> units at <span class="math-container">$A$</span> and a mass of <span class="math-container">$1$</span> units at <spa... |
4,560,406 | <p>I am trying to model the temperature function using the following equation:</p>
<p><span class="math-container">$T(d)=c_0+c_1 \cos (\frac{2\pi}{365} d)$</span></p>
<p>Where <span class="math-container">$d$</span> is the day of the year, and <span class="math-container">$T(d)$</span> is the temperature on that day. I... | ajotatxe | 132,456 | <p>Using analytic geometry, we can put WLOG <span class="math-container">$A=(-1,0)$</span>, <span class="math-container">$C=(2,0)$</span>, <span class="math-container">$B=(3t,3s)$</span> (where <span class="math-container">$s\neq 0$</span>), <span class="math-container">$D=(0,0)$</span>. Let <span class="math-container... |
2,004,745 | <blockquote>
<p>Find point on the curve $x^2+2y^2=6$ whose distance from the line $x+y-7=0$ is, minimum</p>
</blockquote>
<p>$\bf{My\; Try::}$ Let $(x,y)$ be any point on the curve $x^2+2y^2=6\;,$ Then we have to</p>
<p>minimize $\displaystyle \left|\frac{x+y-7}{\sqrt{2}}\right|$</p>
<p>Using $\bf{Cauchy\; Schwarz... | Samrat Mukhopadhyay | 83,973 | <p><strong>Hint:</strong> Use <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow noreferrer">Stirling's approximation</a> $$n!\approx n^n e^{-n}\sqrt{2\pi n}$$ for large $n$.</p>
|
23,704 | <p>I hit a snag whilst revising some log rules, could anyone confirm my suspicion:</p>
<p>$$\log _b \left( x \right) = \log _b \left( y \right) \rightarrow x = y ?$$</p>
| lhf | 589 | <p>By definition, $b^{\log_b t} = t$.</p>
|
3,086,979 | <p>Attempt:
I've used that <span class="math-container">$\sin(x) = (2t)/(1+t^2)$</span> and <span class="math-container">$\cos(x) = (1-t^2)/(1+t^2)$</span>.
However I don't seem to get a perfect square, instead I get <span class="math-container">$$(2/((1+t^2)^2))(14t^2 +20t + 38)$$</span>.</p>
<p>I'm not sure if there... | Fred | 380,717 | <p>If <span class="math-container">$x= \pi/4$</span>, then compute that </p>
<p><span class="math-container">$2(1+ \cos x)(5 \sin x+ 12 \cos x+13)$</span> is not a natural number !</p>
|
1,684,045 | <p>Let $0<2x<\pi$. If $$\sum_{n=-2}^{\infty}\cos^n x=8,$$ then please find $x.$ </p>
<p>I tried $\sum_{n=-2}^{\infty}\cos^n x=\frac{1}{1-\cos x}+\frac{1}{\cos x}+\frac{1}{\cos^2 x}=\frac{1+\cos x}{\cos^2 x \sin^2 x}=8.$ But I cant find $x.$</p>
| Wojciech Karwacki | 242,866 | <p>HINT: consider the series $\sum_{n=-2}^\infty y^n$ with substitution $y=cosx$ and try to find $y$ s.t. $\sum_{n=-2}^\infty y^n = 8$.</p>
|
1,684,045 | <p>Let $0<2x<\pi$. If $$\sum_{n=-2}^{\infty}\cos^n x=8,$$ then please find $x.$ </p>
<p>I tried $\sum_{n=-2}^{\infty}\cos^n x=\frac{1}{1-\cos x}+\frac{1}{\cos x}+\frac{1}{\cos^2 x}=\frac{1+\cos x}{\cos^2 x \sin^2 x}=8.$ But I cant find $x.$</p>
| egreg | 62,967 | <p>Since $0<x<\pi/2$, you know that $0<\cos x<1$, so the series
$$
\sum_{n=-2}^{\infty}(\cos x)^n=\sum_{m=0}^{\infty}(\cos x)^{m-2}=
\frac{1}{\cos^2x}\sum_{m=0}^{\infty}(\cos x)^{m}
$$
converges to
$$
\frac{1}{\cos^2x}\frac{1}{1-\cos x}
$$
Solving
$$
\frac{1}{t^2(1-t)}=8
$$
shouldn't be difficult as it tran... |
51,301 | <p>With <code>n = 17</code> I would like to get </p>
<blockquote>
<p>{4, 1}</p>
</blockquote>
<p>and with <code>n = 999</code></p>
<blockquote>
<p>{31, 6, 1, 1}</p>
</blockquote>
<p>so that, for example,</p>
<pre><code>Total[#^2 & /@ {31, 6, 1, 1}]
</code></pre>
<p>gives</p>
<blockquote>
<p>999</p>
</b... | Sektor | 5,470 | <p>Shamelessly stealing, but using a built-in function</p>
<pre><code>Block[{n = 1, results},
While[(results = PowersRepresentations[#, n, 2]) == {}, n++];
results] &[999]
</code></pre>
|
2,824,996 | <p>I'm investigating iterations using Newton's method for $f(x)=x^5-x+1$.</p>
<p>I'm getting unusual results though. I've found that some starting values will result with a "triple oscillation" of results where iterations will loop between values of approximately $1.0003$, $0.7504$ and $-0.0871$. </p>
<p>However the ... | Joffan | 206,402 | <p>You have $x = 100 \ln x$, which can be written $\frac{\ln x}{x} = 0.01$</p>
<p>Since this question is marked "precalculus", you will have to rely on qualitative arguments to see that $\frac{\ln x}{x}$ will be postive for $x>0$, and tend monotonically towards zero for $x\to \infty$. Since $\frac{\ln x}{x} =0$ a... |
1,566,097 | <p>Prove that the function defined by </p>
<p>$$f(x)=\sin\left(\frac{1}{x}\right)$$</p>
<p>is not uniformly continuous on the interval $(0,1)$.</p>
| Geometry | 295,970 | <p>We need to show:</p>
<p>$$(\exists \epsilon >0) \space \space (\forall \delta >0) \space \space (\exists \space x, y \in \mathbb{R})$$ with $$\left |\space x-y\space \right|<\delta,\space\space\space\space \left|\space f(x)-f(y) \space\right|\geq\epsilon$$</p>
<p>Let $\epsilon=1$. Let $\delta>0$ be arb... |
4,451,700 | <blockquote>
<p>Calculate the line integral
<span class="math-container">$$
\int_{L} \frac{-y \,d x+x \,d y}{x^{2}+y^{2}}
$$</span>
where <span class="math-container">$L$</span> is the line segment from <span class="math-container">$(1,0)$</span> to <span class="math-container">$(0,1)$</span> parametrized by
<span clas... | Andrei | 331,661 | <p>Let's write your integral as <span class="math-container">$$\int_L\vec f d\vec r$$</span>
In Cartesian coordinates <span class="math-container">$$d\vec r=\hat x dx+\hat y dy$$</span> so then <span class="math-container">$$\vec f=\frac{-y}{x^2+y^2}\hat x+\frac x{x^2+y^2}\hat y$$</span>
In polar coordinates <span cla... |
1,631 | <p>What's a good way to format a citation to a MathOverflow post in a paper?
Here is a first draft (with hyperlink, and assuming it's reference number 16, say), but maybe people can think of improvements:</p>
<blockquote>
<p>[16] Pietro Majer, answer of April 8, 2014, 17:26pm HST to <em>MathOverflow</em> question <... | François G. Dorais | 2,000 | <p>On the main site, the (<a href="https://meta.mathoverflow.net/questions/123/can-we-make-the-cite-link-bigger">nearly invisible</a>) cite button found in the share menu gives an example citation and gives both bibtex and amsref <code>misc</code> entries that are easy to cut and paste.</p>
<p><img src="https://i.stac... |
3,789,885 | <p>I need help proving the following trig inequality:</p>
<blockquote>
<p><span class="math-container">$$\cos a-\cos b-\cos c\geq -\frac{3}{2}$$</span> where <span class="math-container">$a+b+c=2\pi$</span> and <span class="math-container">$a,b,c>0$</span>.</p>
</blockquote>
<p>I've found that equality occurs when <... | Michael Rozenberg | 190,319 | <p>We need to prove that <span class="math-container">$$\cos{a}-2\cos\frac{b+c}{2}\cos\frac{b-c}{2}+\frac{3}{2}\geq0$$</span> or
<span class="math-container">$$2\cos^2\frac{a}{2}-1+2\cos\frac{a}{2}\cos\frac{b-c}{2}+\frac{3}{2}\geq0$$</span> or
<span class="math-container">$$4\cos^2\frac{a}{2}+4\cos\frac{a}{2}\cos\frac{... |
2,873,255 | <p>So when deriving the equation for charge, I reached a point where</p>
<p>$\frac{z\eta}{4\epsilon} \sum_{i=1}^N \frac{\Delta (R_i^2)}{(z^2+R_i^2)^\frac{3}{2}}$</p>
<p>But my teacher explains that </p>
<p>$\Delta (R_i^2)$
can be rearranged to
$2R_i \Delta (R_i)$ </p>
<p>but he cannot explain why.
He talks about t... | actinidia | 467,139 | <p>When taking differentials, $$x = f(t) \implies \mathrm dx =
\frac{\mathrm df}{\mathrm dt} \, \mathrm dt$$</p>
<p>Let's apply the above statement. Let $u_i = R_i^2$. As $\Delta u_i$ approaches $0$, it becomes a differential $\mathrm du_i$. So we have</p>
<p>$$u_i = f(R_i) = R_i^2 \implies \mathrm du_i =
\frac{\ma... |
2,873,255 | <p>So when deriving the equation for charge, I reached a point where</p>
<p>$\frac{z\eta}{4\epsilon} \sum_{i=1}^N \frac{\Delta (R_i^2)}{(z^2+R_i^2)^\frac{3}{2}}$</p>
<p>But my teacher explains that </p>
<p>$\Delta (R_i^2)$
can be rearranged to
$2R_i \Delta (R_i)$ </p>
<p>but he cannot explain why.
He talks about t... | WW1 | 88,679 | <p>Unpacking the meaning of $\Delta$ as ...
$$ \Delta f(t)\equiv f(t+\Delta t)-f(t) $$
We can write ...
$$ \begin{array} \\ \Delta x^2(t)
\\=x^2(t+\Delta t) - x^2(t) \\=(x(t+\Delta t)+x(t))(x(t+\Delta t)-x(t))
\\=\left [\Delta x(t) +2x(t) \right ] \Delta x(t) \end{array} $$</p>
<p>So provided that $\Delta x(t) &l... |
2,316,246 | <p>I was pondering on how to answer this question. It looks like this:</p>
<p>In a certain region of the country it is known from past experience that the probability of selecting an adult over 40 years of age with cancer is 0.05. If the probability of a doctor correctly diagnosing a person with cancer as having the d... | Ross Millikan | 1,827 | <p>The $\lambda$ in the <a href="https://en.wikipedia.org/wiki/Poisson_distribution" rel="nofollow noreferrer">Poisson distribution</a> is the expected number, which here is $0.05 \cdot 5000=250$, so it should be $\dfrac {250^{200}e^{-250}}{200!}$</p>
|
2,316,246 | <p>I was pondering on how to answer this question. It looks like this:</p>
<p>In a certain region of the country it is known from past experience that the probability of selecting an adult over 40 years of age with cancer is 0.05. If the probability of a doctor correctly diagnosing a person with cancer as having the d... | knrumsey | 371,419 | <h2>The Binomial Distribution is a better fit here.</h2>
<p>Recall that Poisson has support on all non-negative integers. Since the maximum value for <span class="math-container">$X$</span> is <span class="math-container">$5000$</span>, we know that the Poisson distribution is (technically) inappropriate.</p>
<hr />
<h... |
1,949,133 | <p>$\newcommand{\Reals}{\mathbb{R}}\newcommand{\Vec}{\mathcal{T}}\newcommand{\Brak}[1]{\left\langle #1\right\rangle}$
Just a question about the following exercise.</p>
<p><strong>Exercise</strong></p>
<p>Let $C=\{(x,y,z)\in\mathbb{R}^3|x^2+y^2=1\}\subset\mathbb{R}^3$. The cylinder $C$ can be identified with $M=\mathb... | HK Lee | 37,116 | <p>As a set, $C$ is same to flat cylinder we already know In further in problem $C$ has induced metric from canonical metric on $\mathbb{R}^3$ So it is in fact flat cylinder </p>
<p>And since we give a new coordination which is usually not used we obtain nontrivial geodesic equation </p>
|
886,337 | <p>I need to find $$\int x\sqrt{(a^2 - x^2)}dx$$</p>
<p>I tried putting $x=a cos(t)$ but I ended up getting a very complicated expression, so any tips?</p>
| Gamma Function | 54,750 | <p><strong>Hint</strong> Let $u = a^2-x^2$. Then $du = -2x dx$...</p>
|
2,344,414 | <p>Can anyone tell me how I should calculate this problem?</p>
<p>You are on vacation in a foreign country, and you need help. There are 50 people nearby, and you know <strong>25% of the population of this country speaks English</strong>. </p>
<p>If you were to call for help: </p>
<p>(a) <strong>What is the probabil... | Cornman | 439,383 | <p>This is a Bernoulli-experiment.</p>
<p>When $X$ counts the people which do not speak english, you calculate the propability by:</p>
<blockquote>
<p>$$P(X=0)=\binom{50}{0}\cdot (\frac{1}{4})^0\cdot (\frac{3}{4})^{50}$$</p>
</blockquote>
|
1,552,992 | <p>Let $G$ be abelian. Let $n \in \mathbb{Z}^+$ such that the order of $G$ and $n$ are relatively prime. Show that the function $\varphi : G \rightarrow G$ defined by $\varphi (a) = a^n $ is an automorphism for $a \in G$.</p>
<p>I'm stuck in proving 1-1-ness of the function. Let $a,b$ in $G$ such that $\varphi (x) = \... | Zelos Malum | 197,853 | <p>You can either show that $\phi(x)=\phi(y)$ implies $x=y$ or if $x\neq y$ then $\phi(x)\neq\phi(y)$.</p>
<p>Let's use the latter, if $x\neq y$ then we have that $xy^{-1}\neq e$. This gives us that
$$\phi(xy^{-1})=\phi(x)\phi(y^{-1})=\phi(x)\phi(y)^{-1}=x^n (y^{n})^{-1}$$
for $x^n (y^{n})^{-1}=e$ we must have that $x... |
4,583,108 | <p>Let <span class="math-container">$M=\langle S\mid R\rangle$</span> be a presentation for a monoid. <span class="math-container">$R=\{a_1=b_1,...\}$</span> is a collection of equalities, and <span class="math-container">$M$</span> is defined as the set of equivalence classes under the the reflexive and transitive clo... | Zongshu Wu | 1,080,937 | <p>After thinking for quite some time, I solved my question. For <span class="math-container">$w_1,w_2\in S^*$</span>, write <span class="math-container">$w_1\sim w_2$</span> if it is possible to rewrite <span class="math-container">$w_1$</span> as <span class="math-container">$w_2$</span>.</p>
<p><em>Proof.</em> For a... |
597,638 | <p>I need to evaluate this integral using calculus of residues: </p>
<p>$$\int_0^\infty\frac{\ln(x)}{1+x^4}\mathrm{d}x$$</p>
<p>I know I need to consider $\displaystyle \int_0^\infty$$\frac{\ln(z)}{1+z^4}\mathrm{d}z$.<br>
Then the integrand has singularities at at $-e^{i\pi/4}$, $e^{i\pi/4}$, $-e^{3i\pi/4}$, and $e^{... | Ron Gordon | 53,268 | <p>There are a number of ways to attack this using the residue theorem. One way is to consider the following contour integral:</p>
<p>$$\oint_C dz \frac{\log{z}}{1+z^4} $$</p>
<p>where $C$ is a quarter circle of radius $R$ in the upper-right quadrant, modified by a small quarter-circle of radius $\epsilon$ so as to ... |
890,796 | <p>No homework tag because it is just practice for a final, not for marks:</p>
<p>$\text{Let $S, T \subseteq U$. If $S \bigcap T= \emptyset$, then $S$ and $T$ :}$</p>
<p>A) are always complements of each other in $U$,</p>
<p>B) are mutually disjoint,</p>
<p>C) have no elements in common,</p>
<p>D) choices B and C... | barak manos | 131,263 | <p>Compute the probability of the opposite event, then subtract the result from $1$:</p>
<ul>
<li><p>The number of ways to choose $6$ pairs out of $8$ pairs is $\dbinom{8}{6}=28$</p></li>
<li><p>The number of ways to choose one sock from each one of $6$ pairs is $2^6=64$</p></li>
<li><p>The number of ways to choose $6... |
890,796 | <p>No homework tag because it is just practice for a final, not for marks:</p>
<p>$\text{Let $S, T \subseteq U$. If $S \bigcap T= \emptyset$, then $S$ and $T$ :}$</p>
<p>A) are always complements of each other in $U$,</p>
<p>B) are mutually disjoint,</p>
<p>C) have no elements in common,</p>
<p>D) choices B and C... | Christian Blatter | 1,303 | <p>Let's compute the probability $q$ that you don't get a matching pair.</p>
<p>We are picking the socks one after the other. After the first sock has been picked there remain $15$ socks, and $14$ of these are "allowed". After two picks with no matching pair there remain $14$ socks, and $12$ of these are "allowed", an... |
1,514,028 | <p>It's pretty obvious to me as I see the plot of the two functions but how can I prove it with some algebra?</p>
| Beni Bogosel | 7,327 | <p>(image taken from robjohn's answer <a href="https://math.stackexchange.com/a/75151/7327">here</a>)</p>
<p>A geometrical proof can be seen in the following image:</p>
<p>$\sin x$ is the height from $C$ which is smaller than $BC$, which, in turn is smaller than the length of the arc $BC$, which is equal to $x$.</p>
... |
1,981,806 | <p>$f(x) = \frac{x}{x^2 + 1}$. How do you find $f^{-1}$?</p>
<p>Is try and solve for y and I get the following:</p>
<p>$y = \frac{x}{x^2 + 1}$</p>
<p>$y(x^2 + 1) = x$</p>
<p>$yx^2 + y = x$</p>
<p>$y = -yx^2 + x$</p>
<p>$y = x(-yx + 1)$</p>
<p>It seems whatever I do I can't isolate the x variable, I just go in c... | Piquito | 219,998 | <p>You have easily (<em>solving a quadratic equation</em>) for all $a$ such that $0\lt a\lt \frac 12$, the equality
$$f\left(\frac{1+\sqrt{1-4a^2}}{2a}\right)=f\left(\frac{1-\sqrt{1-4a^2}}{2a}\right)=a$$ This shows that $f$ is not injective so not bijective and has no inverse.</p>
|
4,149,316 | <p>Is <span class="math-container">$f(x,y) = x^2 - 2e^y$</span> concave or convex?</p>
<p>My thought process is as such:</p>
<p><span class="math-container">$f'(x,y)=2x - 2e^y$</span>;</p>
<p><span class="math-container">$f''(x,y)=2 - 2e^y$</span></p>
<p>Thus when <span class="math-container">$y<0$</span> (<span cla... | Eike Schulte | 11,853 | <p>There are two things going on in your question: The first is that your calculation of derivatives for multivariable functions does not make sense; this is explained in the other answers.</p>
<p>The second thing is the question whether a function can be convex or concave only for certain parts of its domain. To illus... |
1,848,321 | <blockquote>
<p>Show that the improper integral $\int_0^1 \left(\left\lfloor\frac{\alpha}{x}\right\rfloor-\alpha\left\lfloor\frac{1}{x}\right\rfloor\right)\mathrm dx=\alpha \ln\alpha$, for $\alpha\in(0,1)$.</p>
</blockquote>
<p><strong>This is an integral of Riemann</strong>. My work:</p>
<ol>
<li>The set of discon... | RRL | 148,510 | <p>The <strong><em>improper</em></strong> integral can be evaluated as the limit of an integral over $[1/n,1]$ as $n \to \infty.$</p>
<p>Making the change of variables $u = 1/x,$ we get </p>
<p>$$\alpha\int_{1/n}^1 \left\lfloor\frac{1}{x}\right\rfloor dx = \alpha\int_{1}^{n } \frac{\left\lfloor u\right\rfloor}{u^2} \... |
1,848,321 | <blockquote>
<p>Show that the improper integral $\int_0^1 \left(\left\lfloor\frac{\alpha}{x}\right\rfloor-\alpha\left\lfloor\frac{1}{x}\right\rfloor\right)\mathrm dx=\alpha \ln\alpha$, for $\alpha\in(0,1)$.</p>
</blockquote>
<p><strong>This is an integral of Riemann</strong>. My work:</p>
<ol>
<li>The set of discon... | tired | 101,233 | <p>I would follow a different route:
First let us change variables $x\rightarrow1/y$ so the problem is equally stated as </p>
<blockquote>
<p>$$
I(a)=\int_1^{\infty}dy\frac{1}{y^2}\left(\left\lfloor a y\right\rfloor- a\left\lfloor y\right\rfloor\right)
$$</p>
</blockquote>
<p>there is a well known <a href="https:/... |
2,387,402 | <p>Is it possible in n dimensions that 3 vectors have a 45-degree angle with each other?</p>
<pre><code><(x,y)= <(x,z) = <(z,y) = 45
</code></pre>
<p>is there also any condition for it? </p>
| angryavian | 43,949 | <p>Three dimensions suffices.</p>
<p>\begin{align}
x &= (1,0,0)
\\
y &=
%(\cos(\pi/4), \sin(\pi/4), 0) =
(1/\sqrt{2}, 1/\sqrt{2}, 0)
\\
z &= (1/\sqrt{2}, 1-1/\sqrt{2}, \sqrt{\sqrt{2}-1})
\end{align}</p>
<p>These are unit vectors satisfying $\langle x,y\rangle = \langle x,z \rangle = \langle y,z \rangle = ... |
2,387,402 | <p>Is it possible in n dimensions that 3 vectors have a 45-degree angle with each other?</p>
<pre><code><(x,y)= <(x,z) = <(z,y) = 45
</code></pre>
<p>is there also any condition for it? </p>
| Ben Grossmann | 81,360 | <p>It's possible whenever $n \geq 3$. It's pretty clear that this isn't possible for $n = 1,2$.</p>
<p>Let $\langle x,y \rangle$ denote the "dot-product" of the vectors $x$ and $y$. Then the angle between $x$ and $y$ is $45^\circ$ if and only if $\langle x,y \rangle/(\|x\| \cdot \|y\|) = \cos(45^\circ) = 1/\sqrt{2}$... |
1,977,081 | <p>You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and con... | Litteul | 380,373 | <p>There is unfortunately many mistakes in the calculation of the expectation proposed:</p>
<p>1) The expectation does not take into account the optionality of the player deciding to play or not.</p>
<p>2) Also since the number of throws is not defined in your calculation and amalgamates all the possible games played... |
1,977,081 | <p>You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and con... | A.G. | 115,996 | <p>You should stop when the expected score from another toss becomes negative or zero. Let $X$ be the score; the expected gain for the next toss is:
$$
1/6\times1+1/6\times2+1/6\times3+1/6\times4+1/6\times5+1/6\times(-X)=\frac{15-X}{6}.
$$
Therefore you should stop as soon as your total score is 14 or more.</p>
<p>Of ... |
98,384 | <p>Hello to everyone.</p>
<p>What the question means is that different ways of
expressing the same relation between the data and unknown variables produce
really weird fit results:</p>
<p>The problem:
I have the unknown variables $x_1,x_2,x_3,x_4,x_5,x_6$
and the observations $\vec{q}_i=\Big(q_i(1),q_i(2),q_i(3)\Big... | Felix Goldberg | 22,051 | <p>The matrix is assumed to be real, correct?</p>
<p>The answer seems to be trivially yes. Indeed, assume that $\lambda_{k} \geq 0$ and $\lambda_{k+1}<0$ and $r \geq k+1$. Then $\sum_{i=1}^{r}{\max(0,\lambda_{i})}=\sum_{i=1}^{k}{\lambda_{i}}$.</p>
|
98,384 | <p>Hello to everyone.</p>
<p>What the question means is that different ways of
expressing the same relation between the data and unknown variables produce
really weird fit results:</p>
<p>The problem:
I have the unknown variables $x_1,x_2,x_3,x_4,x_5,x_6$
and the observations $\vec{q}_i=\Big(q_i(1),q_i(2),q_i(3)\Big... | Denis Serre | 8,799 | <p>The answer is yes, because your function (let me call it $f$) is the maximum of convex functions. As such, it is convex. The formula :
$$f(X)=\max\left(0,\max_{1\le r\le n}\sum_{j=1}^r\lambda_j(X)\right).$$</p>
|
2,989,607 | <blockquote>
<p>Christina is standing in a rectangular garden. Her distances from the corners of the garden are <span class="math-container">$6$</span> meters, <span class="math-container">$7$</span> meters, <span class="math-container">$9$</span> meters, and <span class="math-container">$d$</span> meters, where <spa... | nimmy | 424,703 | <p>We have a relation using Stewart's theorem for an arbitrary parallelogram as follows "if ABCD is a parallelogram and P be any point in plane then <span class="math-container">$2AM^2 +2CM^2 - 2BM^2 + 2DM^2 $</span> = <span class="math-container">$ (AC^2 - BD^2)$</span> .</p>
<p>For a rectangle AC =BD then by proce... |
405,955 | <p>$$(1 + 195) \pmod 7 \equiv \quad ?$$</p>
<p>How would I get the answer? Because when I divide $196 / 7$ I get $28$ which is not a decimal to multiply by $7$.</p>
| Mark Bennet | 2,906 | <p>$196$ is divisible by $7$, so $196 \equiv 0 \mod 7$</p>
|
405,955 | <p>$$(1 + 195) \pmod 7 \equiv \quad ?$$</p>
<p>How would I get the answer? Because when I divide $196 / 7$ I get $28$ which is not a decimal to multiply by $7$.</p>
| DanielY | 47,681 | <p>195 mod 7 is 6, and 1 mod 7 is 1</p>
<p>which means the sum is 7 mod 7 = 0</p>
<p>you recieve 28 which is 0 mod 7, which is the answer</p>
|
1,884,791 | <p>I'm trying to prove the convergence of </p>
<p>$$\int^{\infty}_{0} \frac{1}{e^x+1}dx$$</p>
<p>Any suggestion or hint for how to deal with it?</p>
<p>Thanks in advance!</p>
| Jacob Wakem | 117,290 | <p>It is sufficient to show that the integral from $0$ to $\infty$ of $\exp (-x)$ converges. But this is merely that the integral from - $\infty$ to $0$ of $\exp(x) $converges. That is, $\exp(x)$ from $-\infty$ to $0$ converges, which is just $1-0=1$.</p>
|
2,069,201 | <p>let $a,b,c\ge 0$ and such $a+b+c=3$ show that
$$(a^3+1)(b^3+1)(c^3+1)\ge 8$$</p>
<p><strong>My research</strong>:It seem use Holder inequality,so
$$(a^3+1)(b^3+1)(c^3+1)\ge (abc+1)^3$$
Use AM-GM
$$abc\le\dfrac{(a+b+c)^3}{27}=1$$
what? then I think this method is wrong</p>
| Michael Rozenberg | 190,319 | <p>$\sum\limits_{cyc}\left(\ln(a^3+1)-\ln2\right)=\sum\limits_{cyc}\left(\ln(a^3+1)-\ln2-\frac{3}{2}(a-1)\right)$.</p>
<p>Let $f(x)=\ln(x^3+1)-\ln2-\frac{3}{2}(x-1)$.</p>
<p>Easy to see that $\min\limits_{[0,2]}f=0$.</p>
<p>Indeed, $f'(x)=\frac{3x^2}{x^3+1}-\frac{3}{2}=\frac{3(1-x)(x^2-x-1)}{2(x^3+1)}$ and since $x_... |
2,483,254 | <p>I have to compute the limit</p>
<p>$$
\lim\limits_{N \to +\infty} \sqrt{N+1} \log \left(1+\frac{x}{N+1}\right)
$$
where $x \ge 0$ is fixed. I tried to see the previous as</p>
<p>$$
\log \lim\limits_{N \to +\infty} \left(1+\frac{x}{N+1}\right)^{\sqrt{N+1}}
$$
and to change variable, but it doesn't work. Intuitively... | jonsno | 310,635 | <p>If we are allowed to use <strong>L' Hospital</strong>:</p>
<p>$$\lim_{n\to \infty}\frac{\ln(1+\frac{x}{n+1})}{\frac{1}{\sqrt{n+1}}} = \lim_{n\to \infty}\frac{2(n+1)^{5/2}}{(n+1+x)(n+1)^2} \to 0$$</p>
|
3,151,329 | <p>How to find range of this function, it looks easy but somehow I am not able to do that . <br> What is more important to me is , why is wolframalpha unable to find the range , is it not a simple enough function ?
<img src="https://i.stack.imgur.com/pwGkm.png" alt="enter image description here"></p>
| MachineLearner | 647,466 | <p><span class="math-container">$1-x^2=(1-x)(1+x)$</span> needs to be positive because it is under the root. Hence,</p>
<p><span class="math-container">$$(1-x)(1+x)\geq 0.$$</span></p>
<p>What can you conclude for possible values of <span class="math-container">$x$</span>? The possible values of <span class="math-con... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.